Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds

The metric space of geodesic laminations on a surface II: small surfaces

We continue our investigation of the space of geodesic laminations on a surface, endowed with the Hausdorff topology. We determine the topology of this space for the once-punctured torus and the 4-times-punctured sphere. For these two surfaces, we also compute the Hausdorff dimension of the space of geodesic laminations, when it is endowed with the natural metric which, for small distances, is -1 over the logarithm of the Hausdorff metric. The key ingredient is an estimate of the Hausdorff metric between two simple closed geodesics in terms of their respective slopes.

几乎所有维数下无边光滑流形的微分同胚群或同胚群上的紧开拓扑是极小的

The compact-open topology on the diffeomorphism or homeomorphism group of a smooth manifold without boundary is minimal in almost all dimensions

该研究证明维数非3的连通光滑流形微分同胚群上紧开拓扑是极小的，进而推得维数非3、4的流形同胚群上该拓扑极小，闭流形等满足自动连续性的情形下它还是唯一可分Hausdorff群拓扑。

Quotient topology on the set of commensurability classes of hyperbolic 3-manifolds

We investigate relation between Dehn fillings and commensurability of hyperbolic 3-manifolds. The set consisting of the commensurability classes of hyperbolic 3-manifolds admits the quotient topology induced by the geometric topology. We show that this quotient space satisfies some separation axioms. Roughly speaking, this means that commensurablity classes are sparsely distributed in the space consisting of the hyperbolic 3-manifolds.

Geometry and topology of complex hyperbolic and CR-manifolds

We study geometry, topology and deformation spaces of noncompact complex hyperbolic manifolds (geometrically finite, with variable negative curvature), whose properties make them surprisingly different from real hyperbolic manifolds with constant negative curvature. This study uses an interaction between Kähler geometry of the complex hyperbolic space and the contact structure at its infinity (the one-point compactification of the Heisenberg group), in particular an established structural theorem for discrete group actions on nilpotent Lie groups.

Partially hyperbolic dynamics and 3-manifold topology

This is an expository note intended to illustrate current research in topological study of partially hyperbolic diffeomorphisms in dimension 3 with a beautiful result due to Margulis and Plante-Thurston on topological obstructions for a manifold to admit an Anosov flow.

The compact-open topology on the homeomorphism group of a surface without boundary is minimal

We show that the homeomorphism group of a surface without boundary does not admit a Hausdorff group topology strictly coarser than the compact-open topology. In combination with known automatic continuity results, this implies that the compact-open topology is the unique Hausdorff separable group topology on the group if the surface is closed or the complement in a closed surface of either a finite set or the union of a finite set and a Cantor set.

Local topology in deformation spaces of hyperbolic 3-manifolds

We prove that the deformation space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M with incompressible boundary is locally connected at minimally parabolic points. Moreover, spaces of Kleinian surface groups are locally connected at quasiconformally rigid points. Similar results are obtained for deformation spaces of acylindrical 3-manifolds and Bers slices.

A Fréchet topology on measured laminations and Earthquakes in the hyperbolic plane

We prove that the bijective correspondence between the space of bounded measured laminations $ML_b(\mathbb{H})$ and the universal Teichmüller space $T(\mathbb{H})$ given by $λ\mapsto E^λ|_{S^1}$ is a homeomorphism for the Fréchet topology on $ML_b(\mathbb{H})$ and the Teichmüller topology on $T(\mathbb{H})$, where $E^λ$ is an earthquake with earthquake measure $λ$. A corollary is that earthquakes with discrete earthquake measures are dense in $T(\mathbb{H})$. We also establish infinitesimal versions of the above results.

$H$-Surfaces with Arbitrary Topology in Hyperbolic 3-Space

In this paper, we show that any open orientable surface S can be properly embedded in H^3 as a minimizing H-surface for any 0<=H<1. We obtained this result by proving a version of the bridge principle at infinity for H-surfaces. We also show that any open orientable surface S can be nonproperly embedded in H^3 as a minimal surface, too.

Free vs. Locally Free Kleinian Groups

We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension $<1$ are free. On the other hand we construct for any $ε>0$ examples of non-free purely hyperbolic Kleinian groups whose limit set is a Cantor set of Hausdorff dimension $<1+ε$.

Topology of leaves for minimal laminations by non-simply connected hyperbolic surfaces

We give the topological obstructions to be a leaf in a minimal lamination by hyperbolic surfaces whose generic leaf is homeomorphic to a Cantor tree. Then, we show that all allowed topological types can be simultaneously embedded in the same lamination. This result, together with results of Alvarez-Brum-Martínez-Potrie and Blanc, complete the panorama of understanding which topological surfaces can be leaves in minimal hyperbolic surface laminations when the topology of the generic leaf is given. In all cases, all possible topologies can be realized simultaneously.

蜿蜒双曲作用的局部 $C^0$-半刚性

Local $C^0$-semi-rigidity of meandering-hyperbolic actions

研究蜿蜒双曲作用的局部 $C^0$-半刚性问题，通过证明其在该拓扑下的局部半刚性，将稳定性理论从利普希茨 $C^0$ 拓扑扩展到完整 $C^0$ 拓扑，进而从单一原理恢复相关作用及格的 $C^0$-局部半刚性。

3维流形中横截叶状结构的交：Hausdorff叶空间蕴含叶状拟测地线

Intersection of transverse foliations in 3-manifolds: Hausdorff leafspace implies leafwise quasi-geodesic

本文研究闭3维流形中两横截二维叶状结构相交所产生的一维叶状结构，证明其叶状拟测地性质等价于万有覆盖中诱导叶状结构具有 Hausdorff 叶空间。

单模随机双曲流形的拓扑与动力学

Topology and dynamics of unimodular random hyperbolic manifolds

研究单模随机双曲流形端点空间与测地线流动力学关系，证明有两个无限体积端点意味着递归，无限多个则意味着正漂移和熵，还给出适用于确定性双曲流形的暂态准则，方法基于德劳内图和容量概念。